Question: Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{a^2 - 25}{a^2 - 7a + 10}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 - 25}{a^2 - 7a + 10} = \dfrac{(a + 5)(a - 5)}{(a - 2)(a - 5)} $ Notice that the term $(a - 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a - 5)$ gives: $y = \dfrac{a + 5}{a - 2}$ Since we divided by $(a - 5)$, $a \neq 5$. $y = \dfrac{a + 5}{a - 2}; \space a \neq 5$